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Abstract

In many economic and Operations Research (OR) situations the social configuration of the organization, influences the potential possibilities of all the groups of agents. The important group for an agent consists of the leader, the agent himself and all the intermediate agents that exist in the given hierarchy (peer group). In this study, peer group situations where grey uncertainty is included. Peer group grey situations are introduced and related peer group grey games are constructed. The grey Shapley value is specified as a solution concept with a proposition. As an application auction situations are considered. Finally, a conclusion is given.

Introduction

In many economic and Operations Research (OR) situations the social configuration of the organization influences the potential possibilities of all the groups of agents. In some cases, the set of agents is (strictly) hierarchically structured with a unique leader. In other situations, the potential individual economic possibilities interfere with the behavioristic rules induced by the organization structure.

In a strict hierarchy every agent has a relationship with the leader either directly or indirectly with the help of one or more other agents. The important group for an agent consists of the leader, the agent himself and all the intermediate agents that exist in the given hierarchy (peer group). The hierarchy may be described by a rooted directed tree with the leader in the root, each other agent in a distinct node; and the peer group of each agent corresponds to the agents in the unique path connecting the agent with the leader.

This tree uniquely determines the peer group structure: each agent’s peer group corresponds to the agents in the unique path connecting the agent’s node with the root in the tree. Tree-connected peer group situations are introduced as triplets consisting of the set of agents involved, the peer group structure describing the organization’s social configuration and a real-valued vector that gives the potential individual economic possibilities of the agents. Figure 1 illustrates a peer group situation.

Figure 1.

is a tree and the numbers are the players.

To each tree-connected peer group situation it is possible to associate a peer group game, with the agents as players and the characteristic function defined by pooling the individual economic possibilities of those members whose peer groups belong to the coalition. Peer groups are essentially the only coalitions that can generate a non-zero payoff in a peer group game. Some literature related to peer group games are: Arin and Feltkamp (1997), Branzei, Mallozzi, and Tijs (2002, 2010), Deng and Papadimitriou (1994), Gilles, Owen, and van den Brink (1992), Myerson (1977, 1980), and Owen (1986).

Recently, several papers about cooperative games with grey-valued coalitions appeared (Deng, 1985; Fang & Liu, 2003; Kose & Forrest, 2015; Olgun, Alparslan Gok & Ozdemir, 2016; Palancı, Alparslan Gök, Ergün & Weber, 2015; Tijs, 2003; Zhang, Wu & Olson, 2005). In this case a cooperative game is considered with a grey-valued characteristic function, i.e. the worth of a coalition is not a real number, but an element of the closed grey of real numbers. This means that one observes an element between a lower bound and an upper bound of the worth of the considered coalitions.

In the literature, there are several papers about peer group situations by using crisp numbers and interval numbers (Branzei, Mallozzi & Tijs, 2010; Branzei, Fragnelli & Tijs, 2002). As far as we know, peer group situations have not been modeled by using grey numbers. So our study is pioneering work in this area. In this paper, we introduce peer group grey situations and model these situations by using cooperative grey games.